"Find the least integer n such that f(x)..." solved and explained

Answered question

2022-05-09

Find the least integer n such that f(x) is O(x^n) for: (x^3 +x^2 logx)(logx+1)+(17logx+19)(x^3 +2)

Jazz Frenia

Skilled2023-05-06Added 106 answers

To find the least integer

n

such that

f (x)

O (x^{n})

, we need to find the highest power of

x

in the expression for

f (x)

. Let's expand the expression:

f (x) = (x^{3} + x^{2} \log x) (\log x + 1) + (17 \log x + 19) (x^{3} + 2)

f (x) = x^{3} \log x + x^{3} + x^{2} \log^{2} x + x^{2} \log x + 17 x^{3} \log x + 34 + 19 x^{3} + 38 \log x

The highest power of

x

in this expression is

x^{3} \log x

. Therefore, we can write:

f (x) = x^{3} \log x (1 + \frac{1}{\log x}) + O (x^{3})

Since

\frac{1}{\log x}

goes to zero as

x

goes to infinity, we can ignore it in the

O

-notation. Thus, we can write:

f (x) = O (x^{3} \log x)

Therefore, the least integer

n

such that

f (x)

O (x^{n})

n = 4

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